Mathematical Methods for Physics and Engineering : A Comprehensive Guide

3.8 EXERCISES 3.13 Prove thatx^2 m+1−a^2 m+1,wheremis an integer≥1, can be written as x^2 m+1−a^2 m+1=(x−a) ∏m r=1 [ x^2 − 2 axc ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS 3.19 Use de Moivre’s theorem withn= 4 to prove that cos 4θ=8cos^4 θ−8cos^2 θ+1, and ded ...

3.9 HINTS AND ANSWERS 3.27 A closed barrel has as its curved surface the surface obtained by rotating about thex-axis the part o ...

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS 3.27 Show thatds=(coshx/a)dx; curved surface area =πa^2 [8 sinh(b/a)−sinh(2b/a)]− 2 πab ...

4 Series and limits 4.1 Series Many examples exist in the physical sciences of situations where we are presented with asum of te ...

SERIES AND LIMITS some sort of relationship between successive terms. For example, if thenth term of a series is given by un= 1 ...

4.2 SUMMATION OF SERIES 4.2.1 Arithmetic series Anarithmetic serieshas the characteristic that the difference between successive ...

SERIES AND LIMITS For a series with an infinite number of terms and|r|<1, we have limN→∞rN=0, and the sum tends to the limit ...

4.2 SUMMATION OF SERIES
Sum the series S=2+ 5 2 + 8 22 + 11 23 +···. This is an infinite arithmetico-geometric series witha=2,d ...

SERIES AND LIMITS The difference method may be easily extended to evaluate sums in which each term can be expressed in the form ...

4.2 SUMMATION OF SERIES 4.2.5 Series involving natural numbers Series consisting of the natural numbers 1, 2, 3,..., or the squa ...

SERIES AND LIMITS
Sum the series ∑N n=1 (n+1)(n+3). Thenth term in this series is un=(n+1)(n+3)=n^2 +4n+3, and therefore we can ...

4.2 SUMMATION OF SERIES Integrating the RHS by parts we find S(x)/x=x^2 expx− 2 xexpx+2expx+c, where the value of the constant o ...

SERIES AND LIMITS Again using the Maclaurin expansion of expxgiven in subsection 4.6.3, we notice that S(θ) = Re [exp(expiθ)] = ...

4.3 CONVERGENCE OF INFINITE SERIES 4.3.2 Convergence of a series containing only real positive terms As discussed above, in orde ...

SERIES AND LIMITS which is merely the series obtained by settingx= 1 in the Maclaurin expansion of expx (see subsection 4.6.3), ...

4.3 CONVERGENCE OF INFINITE SERIES Ratio comparison test As its name suggests, the ratio comparison test is a combination of the ...

SERIES AND LIMITS
Given that the series ∑∞ n=1^1 /ndiverges, determine whether the following series converges: ∑∞ n=1 4 n^2 −n− ...

4.3 CONVERGENCE OF INFINITE SERIES Using the integral test, we consider lim N→∞ ∫N 1 xp dx= lim N→∞ ( N^1 −p 1 −p ) , and it is ...

SERIES AND LIMITS The divergence of the Riemann zeta series forp≤1 can be seen by first considering the casep= 1. The series is ...